Let there be the function f(alpha)= (L( v_a - v_0 sin alpha))/(v_0 cos alpha) Where L ,v_a and v_o are constants. Determine alpha such that f(alpha) is minimal ?
2 Answers
F(alpha) = f(alpha)/L qquad F'(alpha) = (f'(alpha))/L qquad " etc "
To optimise
F_alpha = v_a/v_o secalphatanalpha - sec^2alpha = 0
Critical points can, therefore, only occur under these rules:
sinalpha = v_o/v_a qquad qquad implies {(v_o lt= v_a),(cos alpha = sqrt(v_a^2 - v_o^2)/v_a),(tanalpha = v_o/sqrt(v_a^2 - v_o^2) ):}
To determine the max or min nature of the Critical Point, grab the second derivative of
With assumption
Accordingly, there is a min when this occurs:
sinalpha = v_o/v_a
This min will be periodic.
I have solved this way. See the answer below: